Strong approximation in the Apollonian group

The Apollonian group is a finitely generated, infinite index subgroup of the orthogonal group OQ(Z) fixing the Descartes quadratic form Q. For nonzero v∈Z⁴ satisfying Q(v)=0, the orbits Pv=Av correspond to Apollonian circle packings in which every circle has integer curvature. In this paper, we specify the reduction of primitive orbits Pv mod any integer d>1. We show that this reduction has a multiplicative structure, and that mod primes p?5 it is the full cone of integer solutions to Q(v)≡0 for v?0. This analysis is an essential ingredient in applications of the affine linear sieve as developed by Bourgain, Gamburd and Sarnak.     

An obstruction to quantizing compact symplectic manifolds

We prove that there are no nontrivial finite-dimensional Lie representations of certain Poisson algebras of polynomials on a compact symplectic manifold. This result is used to establish the existence of a universal obstruction to quantizing a compact symplectic manifold, regardless of the dimensionality of the representation.

     

Transcendental obstructions to weak approximation on general K3 surfaces

We construct an explicit K3 surface over the field of rational numbers that has geometric Picard rank one, and for which there is a transcendental Brauer–Manin obstruction to weak approximation. To do so, we exploit the relationship between polarized K3 surfaces endowed with particular kinds of Brauer classes and cubic fourfolds.     

Odd order Brauer–Manin obstruction on diagonal quartic surfaces

We determine the odd order torsion subgroup of the Brauer group of diagonal quartic surfaces over the field of rational numbers. We show that a non-constant Brauer element of odd order always obstructs weak approximation but never the Hasse principle.     

Distribution of Eigenvalues for the Ensemble of Real Symmetric Toeplitz Matrices

Consider the ensemble of real symmetric Toeplitz matrices whose entries are i.i.d. random variable from a fixed probability distributionpof mean 0,variance 1, and finite moments of all order. The limiting spectral measure (the density of normalized eigenvalues) converges weakly to a new universal distribution with unbounded support, independent of pThis distribution’s moments are almost those of the Gaussian’s, and the deficit may be interpreted in terms of obstructions to Diophantine equations; the unbounded support follows from a nice application of the Central Limit Theorem. With a little more work, we obtain almost sure convergence. An investigation of spacings between adjacent normalized eigenvalues looks Poissonian, and not GOE. A related ensemble (real symmetric palindromic Toeplitz matrices) appears to have no Diophantine obstructions, and the limiting spectral measure’s first nine moments can be shown to agree with those of the Gaussian; this will be considered in greater detail in a future paper.     

Pin c and Lipschitz Structures on Products of Manifolds

The topological condition for the existence of a pin ^c structure on the product of two Riemannian manifoldsis derived and applied to construct examples of manifolds havingthe weaker Lipschitz structure, but no pin ^c structure.An example of a five-dimensional manifold with this property is given;it is pointed out that there are no manifolds of lower dimension withthis property.     

Degenerate Poisson structures in dimension 3

Formal normal forms of degenerate Poisson structures in dimension 3 are described. The main tool of the study is a spectral sequence previously introduced by the author. In particular, this method allows one to obtain a new proof of the linearizability of Poisson structures with semisimple linear part. However, there are nonlinearizable Poisson structures in dimension 3 as well.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 579–592, April, 1998.The author wishes to thank the referee for pointing out reference [3] and for other useful remarks.     

Minor‐order obstructions for the graphs of vertex cover 6

We provide for the first time, a complete list of forbidden minors (obstructions) for the family of graphs with vertex cover 6. This study shows how to limit both the search space of graphs and improve the efficiency of an obstruction checking algorithm when restricted to k–VERTEX COVER graph families. In particular, our upper bounds 2k + 1 (2k + 2) on the maximum number of vertices for connected (disconnected) obstructions are shown to be sharp for all k > 0. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 163–178, 2002